MEAM 520Homogenous Transformations
Katherine J. Kuchenbecker, Ph.D.General Robotics, Automation, Sensing, and Perception Lab (GRASP)
MEAM Department, SEAS, University of Pennsylvania
Lecture 3: September 13, 2012
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Office Hours
• Monday 1-2 pm :: Denise Wong, GRASP Conference Room
• Tuesday 1:30-2:30 pm :: Katherine Kuchenbecker, Towne 224
• Tuesday 2:30-3:30 pm :: Denise Wong, GRASP Conference Room
• Wednesday 5-6 pm :: Philip Dames, GRASP Conference Room
• Thursday 10-11 am :: Philip Dames, GRASP Conference Room
• Thursday 3-4 pm :: Katherine Kuchenbecker, Towne 224
New Course Website
RotationMatrices
Slides created by Jonathan Fiene
Three-Dimensional Rotation Matrix
x0
y0
y1x1
z0
z1
R0
1 =
x1 · x0 y1 · x0 z1 · x0
x1 · y0 y1 · y0 z1 · y0
x1 · z0 y1 · z0 z1 · z0
SO(3)Special Orthogonal group of order 3
Rotation Matrices
Rotation matrices serve three purposes (p. 47 in SHV):
1. Coordinate transformation relating the coordinates of a point p in two different frames
2. Orientation of a transformed coordinate frame with respect to a fixed frame
3. Operator taking a vector and rotating it to yield a new vector in the same coordinate frame
x0
y0
y1x1
z0
z1
x0
y0
y1x1
z0
z1
p
x0
y0
z0
p0
b
p0
a
θ = 90◦
CompositeRotations
Composition of Rotations with Respect to the Current Frame
Composition of Rotations with Respect to the Current Frame
R0
2 = R0
1 R1
2
the result of a successive rotation about the current (intermediate) frame can be found by post-multiplying by the corresponding rotation matrix
Composition of Rotations with Respect to a Fixed Frame
Composition of Rotations with Respect to a Fixed Frame
the result of a successive rotation about a fixed frame can be found by pre-multiplying by the corresponding rotation matrix
R0
2 = R R0
1
RNote that is a rotation about the original frame
You should read the book and think about this.
ParameterizingRotations
Parameterization of Rotations
Euler Angles Roll, Pitch, Yaw Angles Axis/Angle Representation
in three dimensions, no more than 3 values are needed to specify an arbitrary rotation
R0
1 =
x1 · x0 y1 · x0 z1 · x0
x1 · y0 y1 · y0 z1 · y0
x1 · z0 y1 · z0 z1 · z0
which means that the 9-element rotation matrix has at least 6 redundancies
numerous methods have been developed to represent rotation/orientation with less redundancy
Conventions vary, so always check definitions!
Euler Angles
x0
y0
z0
z3
y3
x3
Define a set of three intermediate angles, φ, θ, ψ , to go from 0 → 3
Euler Angles
x0
y0
z0
y1
x1
z1
aboutφ z0step 1: rotate by
φ
θ
Euler Angles
y1
x1
z1
aboutstep 2: rotate by θ y1
x2
y2
z2
Euler Angles
x2
y2
z2
aboutstep 3: rotate by z2ψ
ψ
x3
z3
y3
Euler Angles to Rotation Matrices
R = Rz,φ Ry,θ Rz,ψ
=
cφcθcψ − sφsψ −cφcθsψ − sφcψ cφsθ
sφcθcψ + cφsψ −sφcθsψ + cφcψ sφsθ
−sθcψ sθsψ cθ
(post-multiply using the basic rotation matrices)
=
cφ −sφ 0
sφ cφ 0
0 0 1
cθ 0 sθ
0 1 0
−sθ 0 cθ
cψ −sψ 0
sψ cψ 0
0 0 1
Roll, Pitch, Yaw Angles
defined as a set of three angles about a fixed reference
x0
y0
z0
φroll
θpitch
ψyaw
Roll, Pitch, Yaw Angles to Rotation Matrices
(pre-multiply using the basic rotation matrices)
R = Rz,φ Ry,θ Rx,ψ
=
cφ −sφ 0
sφ cφ 0
0 0 1
cθ 0 sθ
0 1 0
−sθ 0 cθ
1 0 0
0 cψ −sψ
0 sψ cψ
=
cφcθ cφsθsψ − sφcψ sφsψ + cφsθcψ
sφcθ sφsθsψ + cφcψ sφsθcψ − cφsψ
−sθ cθsψ cθcψ
Axis/Angle Representation
rotation by an angle about an axis in space
x0
y0
z0
θ
k
k =
kx
ky
kz
Axis/Angle Representation
any rotation matrix can be represented this way!
x0
y0
z0
θ
k
Talk to the person next to you.Explain one of the three parameterizationapproaches to your partner, then switch.
Talk about the third one together.
z0
x0
y0
φ
θ
ψ
x0
y0
z0
θ
k
x2
y2
z2
ψ
x3
z3
y3
θ
y1
x1
z1
x2
y2
z2
x0
y0
z0
y1
x1
z1
φ
HomogeneousTransformations
Rigid Motion
a rigid motion couples pure translation with pure rotation
x1
y1
z1
o1
x0
y0
z0
o0
d0
1
rigid motions can be expressed as
v0p = R0
1v1p + d0
1
p
v1p
Homogeneous Transforms
H =
[
R d
0 1
]
a homogeneous transformation is a matrix representation of rigid motion, defined as
where is the 3x3 rotation matrix, and is the 3x1 translation vectorR d
H =
nx sx ax dx
ny sy ay dy
nz sz az dz
0 0 0 1
Homogeneous Transformations
the homogeneous representation of a vector is formed by concatenating the original vector with a unit scalar
P =
[
p
1
]
P =
px
py
pz
1
where is the 3x1 vectorp
Homogeneous Transformations
rigid body transformations are accomplished by pre-multiplying by the homogenous transform
P0
= H0
1 P1
{example with quadrotor model}
x0
y0
z0
o0
P0
= H0
1 P1
Rz,θ =
cos θ − sin θ 0
sin θ cos θ 0
0 0 1
x0
y0
z0
o0
Homogeneous Transformations
composition of multiple transforms is the same as for rotation matrices:
H0
2 = H0
1 H1
2
post-multiply when successive rotations are relative to intermediate frames
pre-multiply when successive rotations are relative to the first fixed frame
H0
2 = H H0
1
Homogeneous Transformations
Composition (intermediate frame)
H0
2 = H0
1 H1
2 =
[
R01 d0
1
0 1
] [
R12 d1
2
0 1
]
=
[
R02 R0
1d12 + d0
1
0 1
]
Inverse Transform
H1
0 =
[
R10 d1
0
0 1
]
=
[
(R01)
!−(R0
1)!d0
1
0 1
]
Homework 1
Homework 1:
Rigid Motions and Homogeneous Transformations
MEAM 520, University of PennsylvaniaKatherine J. Kuchenbecker, Ph.D.
September 11, 2012
This assignment is due on Tuesday, September 18, by 5:00 p.m. sharp. You should aim to turn thepaper part in during class that day. If you don’t finish until later in the day, you can turn it in to ProfessorKuchenbecker’s office, Towne 224. The code must be emailed according to the instructions at the end ofthis document. Late submissions of either or both parts will be accepted until 5:00 p.m. on Wednesday, butthey will be penalized by 25%. After that deadline, no further assignments may be submitted.
You may talk with other students about this assignment, ask the teaching team questions, use a calculatorand other tools, and consult outside sources such as the Internet. To help you actually learn the material,what you write down should be your own work, not copied from a peer or a solution manual.
Book Problems (30 points)
The first set of problems is from the textbook, Robot Modeling and Control by Spong, Hutchinson, andVidyasagar (SHV). Please follow the extra clarifications and instructions when provided. Write in pencil,
show your work clearly, box your answers , and staple together all pages of your assignment.
1. SHV 2-10, page 66 – Sequence of Rotations (5 points)Please specify each element of each matrix in symbolic form and show the order in which the matricesshould be multiplied; as stated in the problem, you do not need to perform the matrix multiplication.
2. SHV 2-14, page 67 – Rotating a Coordinate Frame (5 points)Sketch the initial, intermediate, and final frames by reading the text in the problem. Then find R intwo ways: by inspection of your sketch and by calculation. Check your solutions against one another.
3. SHV 2-23, page 68 – Axis/Angle Representation (10 points)
4. SHV 2-39, page 70 – Homogeneous Transformations (10 points)Treat frame o2x2y2z2 as being located at the center of the cube’s bottom surface (as drawn in Figure2.14), not at the center of the cube (as stated in the problem).
MATLAB Programming (30 points)
This class will use MATLAB to analyze and simulate robotic systems and also to control real robots. WhileProfessor Kuchenbecker loves MATLAB, she recognizes that it can be difficult to use at the start. Even ifyou don’t like MATLAB now, please give it a chance, and come to office hours or contact the teaching teamif you feel lost or frustrated.
Your task for this question is to update a provided MATLAB script so that it animates the movement ofrectangular block that was moved in a specific way. The motion was captured on video, and the positionsand orientations of the block were recorded over time using a Ascension TrakStar magnetic motion trackingsystem that includes a sensor located inside the block.
1
x0
z0y0
frame 0 - camera
x1
z1
y1
frame 1 - transmitter
y 2
z 2
x 2 frame 2 - sensor
the flying box
2010
0
100
10
5
0
5
10
15
20
25
z0 (i
n.)
Flying Box by PUT YOUR NAME HERE
x0 (in.)y0 (in.)
Programming Homework Tips:
1. Write out your approachbefore sitting down to program.
2. Start early to give yourself time to figure things out.
Questions ?
